Regularization and robustness appear at first glance to be two distinct ideas with which to encourage a nominal formulation to enjoy better out-of-sample guarantees. However, intimate connections between both ideas have been reported before by [1]}.
In particular, [2]} state an equivalence between Wasserstein DRO formulations and a certain gradient-norm regularization formulation. In Proposition REF , we establish as an aside that the SVP formulation of [3]} enjoys a robust interpretation with respect to a particular ellipsoidal \(\chi ^2\) -divergence ambiguity set. Furthermore, from the previous discussion it is clear that adding robustness reduces to regularization with
\(R(x, \hat{\mathbb {P}}_T, T) = \sup _{\mathbb {P}^{\prime } \in \mathcal {P}} \left\lbrace  \mathbb {E}_{\mathbb {P}^{\prime }}(\ell (x,\xi ))\ : \ \mathbb {P}^{\prime } \in \mathcal {U}_T(\hat{\mathbb {P}}_T) \right\rbrace  - c(x, \hat{\mathbb {P}}_T)) \ge 0.\) 
